Chapter 2.8, Problem 3P

Find the power series for $e^x\cos x$ and for $e^x\sin x$ from the series for $e^z$ in the following way: Write the series for $e^z$; put $z=x+iy$.

Show that $e^z=e^x(\cos y+i\text{}\sin \text{}y)$; take real and imaginary parts of the equation, and put $y=x$.